Bayesian Analysis of Flipper Length

The Species: Chinstrap Penguins

Chinstrap penguins are one of the most abundant penguin species in the Antarctic, with an estimated population of nearly eight million pairs. They are the most abundant penguin species breeding at the South Sandwich Islands. According to the study, the penguins take more than 10,000 naps a day, with each one lasting only four seconds, totaling 11-12 hours per day. Today we will analyze flipper lengths for Chinstrap penguins.

Bayesian Model Selection

Y_i|\mu \sim N(\mu, \sigma^2)

Since flipper lengths are generally well-distributed and approximately normal for penguins, the Normal-Normal Bayesian Model is the most appropriate for the analysis of Chinstrap penguin flipper lengths.

Prior Specification: The Data

Using prior studies we have determined the average flipper length of Chinstrap penguins is 193.37mm with a standard deviation of 6.12 [1]

Prior Specification: The Model

\mu \sim N(193.7, {6.12}^2)

The Sample Data

There are 68 data points for the Chinstrap penguins. The sample mean is 195.8235mm with a standard deviation of 7.131894.

The Posterior Model

Model	Mean	Mode	Variance	SD
Prior	193.37	193.37	37.4544	6.12
Posterior	195.7721	195.7721	0.7329711	0.8561373

Credible Intervals

qnorm(c(0.025, 0.975), 195.77, 0.86)

95\% \text{ CI} = [194.08, 197.46]

Hypotheses

Is the average flipper length for Chinstraps between 215 and 225mm?

Null Hypothesis H_0:
- H_0: \mu \in (215,225)
Alternative Hypothesis H_1:
- H_1: \mu \notin (215,225)

Hypotheses: Conclusions

The 95% credible interval for the posterior is [194.08, 197.46], which does not overlap with the range specified in the null hypothesis [215, 225]. This indicates the posterior belief supports values below 215mm, resulting in no evidence for the null hypothesis.

Posterior Probability

posterior_prob <- pnorm(225, 195.77,0.86)-pnorm(215,195.77,0.86)
posterior_prob

[1] 0

P(215\le\mu\le225|Y=68)=0 This confirms the conclusion drawn previously.

Bayes Factor

Hypotheses	Prior Probability	Posterior Probability
H_0: \mu \in (215,225)	P[H_0] = 0.9998	P[H_0 \mid Y = 68] = 1
H_1: \mu \notin (215,225)	P[H_1] = 0.0002	P[H_1 \mid Y = 68] = 0

Code

prior_prob <- pnorm(225,193.37, 6.12)-pnorm(215,193.37,6.12)
prior_odds <-(prior_prob/(1-prior_prob))
posterior_odds <- (posterior_prob/(1-posterior_prob))
bayes_factor <- posterior_odds/prior_odds

\text{Bayes Factor}= \frac{\text{Posterior Odds}}{\text{Prior Odds}}= \frac{0}{0.0002}=0

BF < 1
- Plausibility of H_1 decreased in light of the observed data
This along with the 95% credible interval confirms our belief that the average flipper length is not between 215mm and 225mm.

Summary

Chinstrap penguins had flipper length
- Prior: \mu \sim N(193.37, 6.12^2)
- Posterior: \mu|(Y=68) \sim N(195.77, 0.86^2)
We noted no extra large penguins in our sample.
- Maximum recorded flipper length was 212.0mm
Bayes Factor and CI indicated that H_0 should be rejected
- Confirming H_1, average flipper length is not between 215mm and 225mm

Summary

Chinstrap penguins had flipper length
- Prior: \mu \sim N(193.37, 6.12^2)
- Posterior: \mu|(Y=68) \sim N(195.77, 0.86^2)
We noted no extra large penguins in our sample.
- Maximum recorded flipper length was 212.0mm
Bayes Factor and CI indicated that H_0 should be rejected
- Confirming H_1, average flipper length is not between 215mm and 225mm

References

[1]

Moreno, Barbosa, León, and Fargallo, “Phenotypic selection on morphology at independence in the chinstrap penguin pygoscelis antarctica,” J. Evol. Biol., vol. 12, no. 3, pp. 507–513, May 1999.

Study Locations

Palmer Station, Deception Island(Prior Study)